Abstract:  We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence (1, 4, 4², 4³, ...) which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number of weighted Motzkin paths and the sequence (1, k, k², k³, ...) for any k ¸ 2. By extending this argument to partial Motzkin paths with multiple elevation lines, we give a combinatorial proof of an identity recently obtained by Cameron and Nkwanta. A matrix identity on colored Dyck paths is also given, leading to a matrix identity for the sequence (1, t² + t, (t² + t)², ... ).

  AMS Classification:  05A15, 05A19.

  Keywords:  Catalan number, SchrÄoder number, Dyck path, Motzkin path, partial Motzkin path, free Motzkin path, weighted Motzkin path, Riordan array.

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